NASA Technical Reports Server (NTRS) 20040090462: A Study of Supersonic Surface Sources: The Ffowcs Williams-Hawkings Equation and the Kirchhoff Formula

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AIAA 98-2375
A Study of Supersonic Surface
Sources-The Ffowcs Williams-Hawkin gs
Equation and the Kirchhoff Formula
F. Farassat and Kenneth S. Brentner
NASA Langley Research Center
Hampton, Virginia
M. H. Dunn
Old Dominion University
Norfolk, Virginia
Corrected September 21,1998
-
~
4th AIANCEAS Aeroacoustics Conference
June 2-4, 1998 / Toulouse, France
For permission to copy or republish, contact the American Institute of Aeronautics and Astronautics
1801 Alexander Bell Drive, Suite 500, Reston, VA 201 91
A STUDY OF SUPERSONIC SURFACE SOURCES-
THE FFOWCS WILLIAMS-HAWKINGS EQUATION AND THE
KIRCHHOFF FORMULA'
F. Farassat" and Kenneth S. Brentner? M.H. Dunnf
NASA Langley Research Center, Old Dominion University,
Hampton, Virginia Norfolk, Virginia
Abstract delta function. The method of solution of these two
equations are, thus, identical.
In this paper we address the mathematical problem
of noise generation from high speed moving surfaces. Obtaining various forms of the solution of these
The problem we are solving is the linear wave equation equations for subsonic surfaces is fairly easy. These
with sources on a moving surface. The Ffowcs Will- solutions (Formulations 1 and 1A of Farassat) have been
iams-Hawkings (FW-H) equation as well as the govern- published el~ewhere~W-e~ .w ill not, therefore, address
ing equation for deriving the Kirchhoff formula for the subsonic case here. We mention that the cominon
moving surfaces are both this type of partial differential forms of the solution for subsonic surfaces have a Dop-
equation. We give a new exact solution of this problem pler singularity which make them unsuitable for super-
here in closed form which is valid for subsonic and sonically moving surfaces. To obtain new forms of
supersonic motion of the surface but it is particularly solution of the FW-H and the K equations for supersonic
suitable for supersonically moving surfaces. This new surfaces, we must integrate the Green's function of the
solution is the simplest of all high speed formulations of wave equation in a different way than the subsonic
Langley and is denoted formulation 4 following the tra-
case'". This was fully recognized by Ffowcs Williams
dition of numbering of our major results for the predic-
and Hawkings' and they laid the foundation for the
tion of the noise of rotating blades. We show that for a
work we present here. The solution of the supersonic
smooth surface moving at supersonic speed, our solu-
problem is considerably more difficult than the subsonic
tion has only removable singularities.Thus it can be
case and it has taken a lot longer to fully overcome the
used for numerical work.
many mathematical obstacles.
1. Introduction
To understand the nature of the complexities
The problem of noise generation from moving bod- involved, one must recognize that the problem as treated
ies is very important in aeroacoustics. Two current here is four dimensional. We are interested in formula-
methods of attacking this problem are the acoustic anal- tions which are suitable for efficient numerical noise
ogy and the Kirchhoff formula for moving surfaces. The prediction from rotating machinery. This requirement
acoustic analogy method is based on different forms of puts a restriction on what forms of the solution of the
the solution of the Ffowcs Williams-Hawkings (FW-H) FW-H and the K equation are acceptable to us. In
equation'. This is a linear wave equation with sources practice, it has been found that the common formula-
on a moving surface. The Kirchhoff formula is also tions for subsonic surfaces are much more efficient than
derived from a linear wave equation with sources on a the supersonic formulations even if the latter can also be
moving surface2. For simplicity, we refer to this wave used for subsonic surfaces. Thus, one is forced to use
equation here as the Kirchhoff (K) equation. Using more than one formulation in any noise prediction code
generalized function theory, both the FW-H and the based on the FW-H and K equations.
K equations can be written with inhomogeneous source
terms involving the Dirac delta function with support on In noise calculation, a moving surface, such as a
the moving surface f = 0 and the first derivatives of this blade, is divided into panels and the noise generated by
each panel is summed up to get the total noise from the
*
Senior Research Scientist, AIAA Associate Fellow surface. This means that the FW-H and K equation must
?Research Scientist, AIAA Member be solved for an open surface, e.g., a panel on the
$Associate Professor, AIAA Member
moving surface. Thus we must solve these two equa-
tions with inhomogeneous source terms that have a
# This paper is declared a work of the United States Government and is
not subject to copyright protection in the United States. Heaviside function multiplying the Dirac delta functions
which describe the open surface. The mathematical
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treatment of these inhomogeneous source terms pro- We next find all the surface contributions of the last
duces an additional complexity in obtaining solutions of term of Eq. (1) by taking the space derivatives explicitly
these equations for the supersonic case. and using the rules of generalized differentiati~n~>~>~.
We get
In the subsonic case, the Doppler factor appears as
the result of integration of the source time variable in the a
Green’s function solution of the wave equation after a 02Pf = ,,cLPun-(P-P~)vnl~(f)}
Lagrangian frame is introduced in which the surface is
1
time independent. This step makes the problem essen-
6 2
tially three dimensional. To get other forms of the solu- + a{[ PVnUi - (P - P0)C n i 1 w
tion suitable for a supersonically moving surface, one
a2
must use integration over the influence surface of the JT.. T~
+ aLxJ n iS (f>+a-x iaxj H(f
observer space-time variables (x, t) which we call the
X surface . This surface is more fundamental to the
- Q1+ Q2 + Q3 + Q4 (2)
solution of the wave equation than the actual moving
surface over which the subsonic formulations are We ~ $ t1he n consider a; open surface described by
integrated. f=O, f > 0 where f = f = 0 is the equation of the
7
edge of this open surface2’6. We define such that
In the next section the governing equations of the V f = v where-v is the unit inward geodesic normal to
problem under consideration are presented. In Section 3, the edge f = f = 02>6.T o calculate the noise from
we will give a new solution of the FW-H and the K this open surface, we must multiply Sf) in Ql, Q2 and
equations in closed form for supersonic surfaces. In Sec- Q3 by the Heaviside function H(f ). We will next use
tion 4 we will show that the singularities of the solution the concept of restriction of a variable to the surface f =
are integrable. The concluding remarks follow. 0 and then take the derivatives of Ql and Q2 terms
explicitly6. We use a tilde under a symbol to signify
2. The Governing Equations restriction.
The Ffowcs Williams-Hawkings Equation and the Introducing the notations
Kirchhoff equation are quite well-known in
aeroacoustics. We will need a special form of these E = Pu,-(P-Po)V, (3a )
equations suitable for our work. The FW-H equation for
2
a moving surface f(x, t) = 0 where f > 0 outside the Ei = pvn uz.- (p-po)c ni, (3b)
body is
we have
a a
02Pf = -a{t[ Pun-(P-P,)vn16(f)} Ql = at [E H(f)W)l
2 a
where pf is (p - po)c , p is the density, and po and c Q2 = [EiH (f)W)l
are the density and speed of sound of undisturbed
medium. The local normal fluid and body velocities are
denoted by u, and v,, respectively. The Lighthill stress = v2 ’ E, H ( f > W + E,ViS(f>W
tensor is denoted Tij and p is the surface pressure on
-2 HfEnn, H(f>W>
f = 0. Note that we assume that the surface f is defined
such that V f = n where n is the unit outward normal + Eini H(fP’(f (5)
to this surface. The Heaviside function is denoted HO.
As proposed by Ffowcs Williams and Hawkings’, the
moving surface f = 0 can be penetrable and we assume
so here.
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Here, v, is the local velocity of the edge along the geo-
desic normal v with components vi, ETi s the projection j Q ( ~ > 6 ( f >=d ~j Q (15)
vector E, on the surface off = 0 and Hfis the local mean f=O
curvaturev o2f the s’.u rfacef= Os,’. The surface divergence where Hf is the mean curvature off = Os,’. Introduce a
ofETi s . ET using Eqs. (4) and (51, we get new generalized function (distribution) Si(f> by the
following relation:
Ql + Q2 + Q3 = q1 H(f>W>+ q2 H(fP’(f)
+q3 S(f>S(f> (6)
where we have defined the following symbols
The subscript s in Si(f> stands for “simple” which
, emphasizes the similarity of Si(f> to the one dimen-
JT, .
q1 = -anxi +lJ+V2.ET-2 HfEn (7) sional S’(x> that behaves as follows:
j
@(x> S’(X> dx = -@’(O) (17)
Now using the results of eqs. (15 ) and (16 ) in Eq. (14),
q3 = P(VnUv - UnVv) + (P - Po)VnVv (9) we see that the following relation holds:
q(f>
In Eq.Eq.Eq. (7), E, = E,n, and M, = v,/c is the local S’(f> = + 2 Hf S(f> (18)
Mach number on f = 0. Note that q2 in Eq. (6) is
Equation (18 ) is next used in Eq. (10 ) which is writ-
restricted to the surfacef= 0. Also note that in Eq. (5),
ten as
we have dropped the restriction on any variable that
8
multiplies S(f, if it is not differentiateahus we write q(f>
and not E . There is also no need to use restriction sign 02P’ = 41 H(f>S (f>+ 42 H(f>
on V2 . ET since ETi s already restricted tof= 0. It is S(f>
important to recognize that 8 is the rate of change of E + q3 S(f> (19)
as measured by an observer on the surface. where now only the definition of q1 is changed as fol-
lows:
The FW-H equation for an open penetrable surface
moving at supersonic speed is:
02P’ = 41 H(f>W>+ q2 H(f>s’(f>
+q3 S(f>S(f> (10)
and
A similar equation is also obtained for derivation of the
Kirchhoff formula2 for an open surfacef= 0, f > 0
where we have:
2
q1 = - w - - M1 a--p-f- (M1 a p-’ )+2 H p‘ (11) + 2HfMn (K eq.1 (21)
an c n at cat -n f
In the following section, we give the full solution of
Eq. (19).
Remark. We will not address the solution of the
q3 = MnMvP’ (13) FW-H equation with the pure quadrupole term alone:
We will go one further step here in preparation of
obtaining the new formulation. We note that for a sur-
facef= 0, lVfl = 1 , we have the following results2’?
The solution using the collapsing sphere approach is
singularity free and is given
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3. Solution of Wave Equation With Sources on A be the determinant of the coefficients of the first funda-
Moving Open Surface mental form in the new variables. Let u l + g , then Eq.
(28) becomes
In this section, we give the solution to the following
three wave equations:
n2+3= q3(x, t>S (f>S(f> (25)
The source terms here are similar to those of Eq. (19 ). 2 3
x6',(u3)du du d2 (29)
The treatment of these equations are discussed in two
references by Farassat2>6W. e will use the solutions to
Note that we have used
eqs. (23) and (25) given in these references but will give
here a new and particularly simple solution of Eq. (24).
The materials presented in these references are essential
in understanding what follows.
Let F(y;X , t> = f(y, t- = [f(y, 2)1vet, ianngd wg(i2th) m Suis t( ub 1e) rewsthriecrtee dt hteo fc u=r v0a btuercea utseerm w eo fa rSe '(due a1l) -
and F(y;x,t ) = [f(y,2 )Ivet , where the sub-
has already been removed (see Eq. (18)) and added to
script ret stands for retarded time. _The influence surface
ql. We must mention here that, in new variables
of the open surface f = 0, f > 0 is called the 1 2
X - surface and is described by F = 0, k > 0 2,6. The g2(y,2 ) = q2[y (u > > 0,2) > 21 f
edge of this surface is the L-curve described by The condition g= 0 in Eq. (29) implies that
F = k = 0. Below, we use (x, t) and ('y, 2) as the u 1 = u 1(u2 , u 3, 2) . We will use this result in the inte-
observer and the source space-time variables, respec- gration of Sl, (ul) in Eq. (29). We get
tively. The solution of Eq. (23) is2>6:
i;"~4 ~ + ~ (2x) ,
- -
j
4 ~ + ~ (t>x =, X[q1~,,,
(26)
where
We have the following results
2 2
A = 1 +M,-2 M, COS^ (27)
Here, M, is the local normal Mach number of the sur-
face f = 0 and cos0 = n . i where n is the unit outward
normal to f = 0, i = (x - y)/ri s the unit radiation vector
from the source to the observer and r = Ix - yl.
We now consider Eq. (24). The formal solution of
this equation using the Green's function method is
1
4 ~ + ~ (t>x =, jig2 2) H(f)6i(f)6(g)d yd~(2 8) (34)
a
where g = 2 - t + r/c. We now introduce a new local au3 = - cot0 v ' tl S(f) (35)
frame (ul, u2, u3) where u3 =f and u1 and u2 are the
Gaussian coordinates on f = constant, extended from where tl is the unit vector along the projection of i on
f = 0 along local normal. We assume that u1 is the length the local tangent plane and Tf,.(sum on i) is the
variable along the projection of i on the local tangent Christoffel symbol of second kind in the new coordinate
plane and u2 is the length variable along i x ii . Let g(2) system. Note that we have a locally orthogonal frame
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which gives g(2)= 1 at the origin but because we have a Therefore, Eq. (36) can be written as follows:
curved surface, we get Eq. (34). Therefore, in the rest of
the algebraic manipulations, we will set g(2)= 1.
We can now write Eq. (31) in the following form cos 0 cos 0 dX
after taking the derivative with respect to u3 of the inte-
1 F=O
grand: ~ F>O
47@2(x, t>
- cos0 cos0 I
- F>O
+ c J- Tlii q2 du2dT
rsin 0 I F=O (44)
We have separated the integrals over the X surface in
-
this equation to simplify the analysis of the singularities.
Finally, the solution of Eq. (25) was also given by
where y is the angle between-? and the edge of the
Farassat2>6a s follows:
open surface described by f = f = 0.
In our previous work, we have used dT for du2
where dT is the element of the curve of intersection of
f = 0 with the collapsing sphere g = 0. We have shown F=O
also that6>10
We have thus given the full solution of Eq. (19) which
cdTdT dX
- (37) we refer to as formulation 4.O nly Eq. (44) in our analy-
sin0 A
sis is new. In the next section, we will discuss the
important question of the singularity of the solution of
Eq. (19).
where L is the edge of the X surface described by 4. A Study of the Singularities of the Solution for
-
F = k = 0 and Supersonic Surfaces
We will now address the question of the singulari-
A. = IVFxVFI = Aisin0' (39) ties in the solution of the FW-H and K equations. There
has been a general belief among the researchers, the
authors of this paper included, that the solution of these
equations lead to nonintegrable singularities for some
observer space-time variables (x, t). We will show here
that for a smooth surface, all the singularities of the new
solution are integrable. We assume that f = 0 is not an
open surface and thus we only consider the integrals
-2 2 over the X - surface : Fb;x , t) =flv, t - r/c) = 0. See
A = 1+M,-2Mvcos8 (42) articles by Farassat, De Bernardis and Myers for the
analysis of the singularities of the line integralsl1>l2.
As will be seen below, the analysis of the singularities
cos0 = v ' ? = v ' tl (431 of the surface integrals is very difficult.
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American Institute of Aeronautics and Astronautics
There are two kinds of singularities in the surface a constant. We next use the relation"
integrals: i) sin0 = 0, i. e., 0 = 0" or 180" but
A f 0,a nd ii) A = 0 . We will show below that A = 0
also implies 0 = 0" or 180" . Condition i) means that at
some source time, the collapsing sphere g = 0 is tangent where r is the curve of intersection of g = 0 with f = 0.
to the surface f = 0 at a point where M, f f1 . Condition Near the point A, above, dT = bdq, sin0 = b/r,
ii) means that at some source time g = 0 is tangent to where b is the radius of r which is a small circle, and q
f = 0 at a point where M, = f1 . We will prove these is the azimuthal angle around r. Thus, we have
assertions for appearance of singularity A = 0 below.
We assume that f = 0 is convex with no saddle points. _&-- crdqd7
(51)
A
Consider a rotating surface part of which moves at
2
supersonic speed. We can write A as follows: which means if go.', 7) is continuous, then
A 2 = (l-MncOsO) 2 +M,2sin. 20 . (46) I, = C j d7j2oR [q,lyetdq (52)
T
Therefore, A = 0 if sin0 = 0 and 1 -M,cos0 = 0
simultaneously. This means that we must have M, = 1 is integrable. Therefore, we have shown the integral
and 0 = 0" or M, = -1 and 0 = 180". The
geometrical interpretation of these conditions is obvi-
ous. Since f = 0 is assumed to be moving supersonically
on part of its surface, there is a curve Y on f = 0 on F>O (53)
which M, = 1 or M, = -1 . If at any source time T ~ , i s integrable.
the collapsing sphere g = 7 - t + r/c = 0 is tangent to
f = 0 at a point on Y , then the integrands of the surface For I,, we use the fact that sin0 = b/r and
integrals of formulation 4 are singular for the time b = C m to write the integral in the form
to = T~ +r/c. Note that, in general, the curve Y is
time dependent but it is not so for a hovering rotor oper-
(54)
ating at supersonic tip speed. This is the case we study
here.
where a = r(7 = 0). This integral is convergent.
We must study the two conditions of appearance of
singularities separately. The reason becomes apparent The study of convergence of I, is very interesting.
below. Let us first write the kinds of integrals we have: It appears that this integral is not convergent. We
manipulate the integrand as follows near the condition
sin0 = 0:
F>O (47)
F>O (48)
We note that tl . V2q2 = -aq2/ab and
cos 0 cos 0
F=O
F>O (49)
We will show that all the singularities are integrable.
and, thus, near the point of tangency off = 0 and g = 0:
Condition i): sin0 = 0, A f 0
Near this point, the intersection of g = 0 and f = 0 is
(57)
a circle of radius b. We can show that if this condition
+
appears at 7 = 0, then as 7 0,b = C m w here C is
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American Institute of Aeronautics and Astronautics
The convergence of I,, therefore, depends on the value shows the envelope of the circles of Eq. (60). We reject
of the following integral when sin0 + 0 : the part of the envelope in the region M < 1 because it
would imply multiple emission from subsonic region
which is impossible. This means that we have no inter-
section of g = 0 and T for 2 < 0. This figure clearly
shows that the X surface looks like a cone near A.
-
We can easily show that
The convergence of I3 is, thus, guaranteed. We conclude
that when sin0 + 0 and A f 0,f ormulation 4 has only
removable singularities.
Condition ii): A = 0 (imDlies sin0 = 0)
As we have shown above, when g = 0 is tangent to
f = 0 at a point where M, = f l , we have A = 0 and
from the tangency condition sin0 = 0. The first thing
we study here is the structure of the X surface near a
-
point where A = 0. We then study the problem of the
singularities of formulation 4.
We consider the condition 0 = 0" and M, = 1
T: Tangent
for a hovering rotor. Figure 1 shows the tangent plane T
to a point on Y looking edgewise at the moment of tan- Plane
gency of the collapsing sphere with f = 0 at the point A.
Note that M, = n = i at this moment so that the Fig. 1. The geometric condition for the appearance of
observer is in the plane shown at the center of the col-
lapsing sphere g = 0. We assume T~ = 0 and r = a at A = 0: 0 = 0", M, = 1,L e., M, = i = n .
the moment of tangency. Since M, = 1 at A, we have
sinp = 1/M where p is the angle that T makes with
Mach number vector M shown in Fig. 1. We now con-
sider the plane T in motion for 121 < E where E > 0 is a p=sinp= l/M
small number. In the frame fixed to T with origin at A as y= cos p /
shown in Fig. 2, the curve of intersection of g = 0 with x-axis out
T, which is a circle is given by the relation
Where Vu = aw , V, = Rw and p and y are defined
in fig. 2. We use R for the distance from A to the axis of
rotation. The center of the circle is at the point
(x,y) = (~V,T-,y VR2) and its radius is V,pl~ .
+
It is clear that as 121 0,t he X - surface looks like a J
vertex of a cone and has no tangent at the vertex coin-
Fig. 2. The coordinate system used to study the intersec-
ciding with A. The condition of A = 0 is thus
equivalent to the X surface becoming pointed and tion of g = 0 with f = 0 near the condition A = 0.
-
having no tangent plane at the point A.
For convergence of I,, the analysis of condition i)
Let us see what the intersection of g = 0 and T will
appear to an observer on the tangent plane T. Figure 3 applies exactly so that I, has removable singularity. The
7
American Institute of Aeronautics and Astronautics
Remark 1. The solutions of the FW-H and the K
y tan a = (R tan p)/a equation here are valid for all range of the surface speed.
Subsonic sin p = l/M But we do not recommend to use the present results for
sin p' cos a subsonic surfaces since much more efficient solutions
for x < 0 =
for numerical method are a~ailable~>~>l'.
Remark 2. It can be shown that had we not added
the surface terms from the quadrupole source term of
the FW-H equation to the thickness and loading source
terms, the resulting solution would be singular when the
condition A = 0 appears. The acoustic pressure signa-
ture will have a logarithmic singularity which will
appear as an infinite pulse. Our analysis shows that
when all the surface sources from thickness, loading and
quadrupole terms are included in the analysis, there is
no infinite singularities in the acoustic field.
5. Concluding Remarks
z-axis out of the plane
We have given the closed form solution of the
FW-H and the K equations for an open surface.
Fig. 3. The shaded area is the trace of the intersection of Although these solutions are valid for all range of Mach
g = 0 in the xy-plane of figure 2 near the condition numbers, we recommend them for the supersonic
A = 0. It is assumed that 2 = 0 for A = 0. motion of the surface because of their complexity. We
have shown that for a smooth surface, the singularities
of the solutions of both of these equations are integrable.
study of the convergence of I2 is different since from The nature of these singularities is explained in this
paper. It is very interesting to note that for the FW-H
Eq. (60) we see that as A+O, we have b = C12
equation, the thickness and loading source terms alone
where C1 is a constant. However, we note that for both
have nonintegrable singularities in the solution. How-
the FW-H and K equations, q2 is proportional to ever, the addition of the surface source terms from the
2 + quadrupole source term removes this singularity. This
Mn- 1. We can easily show that as A 0,
is, of course, expected on intuitive grounds.
Mt- 1 = C22 where C2 is another constant. There- We hope that the present work gives further impe-
fore, the convergence of I2 is guaranteed because near tus to numerical applications of our results in high speed
rotating blade noise prediction. The closed form analytic
A = 0,w e can write I, as
results of this paper open up two other areas of applica-
tion which could help aeroacousticians in their endeavor
to reduce the noise of aeronautical machines. These
areas are: i) qualitative analysis of noise generation
mechanisms by the analytic study of the appropriate
2 integrals in our solutions of the FW-H and K equations,
where qf2 = q2/(Mn- 1). As a matter of fact, I,, is
and ii) approximate analysis of the radiation field from
better behaved for condition ii) than condition i)!
ducted fan inlet and exhaust and other openings that
For I,, the convergence study of condition i) applies radiate sound to an infinite medium. The analysis is sim-
exactly for condition ii). Therefore formulation 4 has ilar to the use of the conventional Kirchhoff formula for
only removable singularities for condition ii) also. the study of diffraction by an aperture.
We conclude that for a smooth surface, the solution Much work is ahead of us in the order of magnitude
of FW-H and the K equations as given here have only analysis of the terms in the solution of the FW-H and the
integrable (removable) singularities for a supersonic K equation. Can a deformable body f = 0 be used to
surface f = 0. The solution of the K equation is, of control noise radiation at high speed? Can we design a
course, known as the Kirchhoff formula for supersonic rigid body with desirable noise radiation property in a
surfaces. given direction?
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American Institute of Aeronautics and Astronautics
References 7. Ram P. Kanwal: Generalized Functions-Theory and
Technique, Second Edition, 1998, Birkhauser,
1. J. E. Ffowcs Williams and D. L. Hawkings: Sound
Boston
Generation by Turbulence and Surfaces in Arbitrary
Motion, Phil. Trans. Roy. SOC.L ondon, Vol. A264, 8. Erwin Kreyszig: Differential geometry, Dover
1969,321-342 Books, 1991
2. F. Farassat: The Kirchhoff Formulas for Moving 9. A. J. McConnell: Applications of Tensor Analysis,
Surfaces in Aeroacoustics- The Subsonic and Super- Dover Books, 1957
sonic Cases, NASA Technical Memorandum
10. F. Farassat: Theory of Noise Generation from Mov-
11 0285, September 1996
ing Bodies With an Application to Helicopter
3. F. Farassat: Linear Acoustic Formulas for Calcula- Rotors, NASA Technical Report R-45 1, 1975
tion of Rotating Blade Noise, AIAA J., Vol. 19,
11. E. De Bernardis and F. Farassat: On the Possibility
1981, 1122-1 130
of Singularities in the Acoustic Field of Supersonic
4. F. Farassat and M. K. Myers: Extension of Kirch- Sources when BEM is Applied to Wave Equation, in
hoff’s Formula to Radiation from Moving Surfaces, Boundary Element Methods in Engineering, B. S.
J. Sound andVib.,Vol. 123, 1988,451-460 Annigeri and K. Tseng (Eds.), Springer-Verlag,
1990, 522-528
5. F. Farassat and G. P. Succi: The Prediction of Heli-
copter Rotor Discrete Frequency Noise, Vertica, Vol. 12. F. Farassat and M. K. Myers: Line Source Singular-
9, 1983,309-320 ity in Wave Equation and Its Removal by Quadru-
pole Sources- A Supersonic Propeller Noise
6. F. Farassat: Introduction to Generalized Functions
Problem, in Structural Acoustics, Scattering and
with Applications in Aerodynamics and Aeroacous-
Propagation, International Conference on Theoreti-
tics, NASA Technical Paper 3428, May 1994, Cor-
cal and Computational Acoustics, Vol. 1, John E.
rected Copy, April 1996
Ffowcs Williams, Ding Lee and Allan Pierce (Eds.),
World Scientific, 1994,29-43
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American Institute of Aeronautics and Astronautics